MPRA
Munich Personal RePEc Archive
A Note on the Size Distribution of
Consumption: More Double Pareto than
Lognormal
Alexis Akira Toda
University of California, San Diego
29 October 2015
Online at
https://mpra.ub.uni-muenchen.de/78979/
A Note on the Size Distribution of Consumption:
More Double Pareto than Lognormal
Alexis Akira Toda
∗October 29, 2015
Abstract
The cross-sectional distribution of consumption is commonly approxi-mated by the lognormal distribution. This note shows that consumption is better described by thedouble Pareto-lognormal distribution (dPlN), which has a lognormal body with two Pareto tails and arises as the station-ary distribution in recently proposed dynamic general equilibrium models. dPlN outperforms other parametric distributions and is often not rejected by goodness-of-fit tests. The analytical tractability and parsimony of dPlN may be convenient for various economic applications.
Keywords: Gibrat’s law, multiplicative idiosyncratic risk, inequality, power law.
JEL codes: D31, E21, G12.
1
Introduction
The cross-sectional consumption distribution is commonly approximated by the lognormal distribution, which is convenient for various economic applications. For instance, a partial list in the working paper version of Battistin et al. (2009)1
reads: (i) lognormality implies that within cohorts, any measure of inequality, such as the Gini coefficient or the Lorenz curve, can be expressed as a function of a single scalar (the variance of log consumption), (ii) it simplifies the handling of possible measurement errors (Banks et al., 1997), and (iii) since Gabaix (1999) shows that the power law2for city populations may arise from an application of
Gibrat’s law of proportionate growth to individual cities, analogous regularities may arise in consumption from Gibrat’s law. Another implication (not listed in the above paper) is that (iv) assuming lognormality of consumption enables us to derive parsimonious analytic expressions for Euler equation aggregation in heterogeneous-agent models (Constantinides and Duffie, 1996; Balduzzi and Yao, 2007). The lognormality in consumption seems to hold within age cohorts (Battistin et al., 2009).
∗Department of Economics, University of California San Diego. 9500 Gilman Drive, La
Jolla, CA 92093-0508, USA. Email: atoda@ucsd.edu. I thank Kieran Walsh for collaboration and discussions, and Thomas Winberry, the associate editor, and two anonymous referees for comments and feedback that significantly improved the paper.
1http://fmwww.bc.edu/EC-P/wp671.pdf
2Usually, “power law” refers to the property that the fraction of units with sizexor larger
is proportional to the power functionx−αwhenxis large enough. The parameterα >0 is
In this note I take a close look at the cross-sectional consumption distribution and examine the third point above: since the cross-sectional distribution of consumption is approximately lognormal within age cohorts (which is consistent with Gibrat’s law), does it obey the power law in the entire cross-section? This question itself has already been answered: Toda and Walsh (2015) show that the entire cross-sectional distribution of consumption in U.S. exhibits a Pareto tail with exponent around 4. Therefore I go one step further by incorporating the recent development in the theoretical literature on inequality. Benhabib et al. (2014) and Toda (2014) theoretically show that, in a certain class of incomplete market dynamic general equilibrium models, the cross-sectional distribution of wealth (and consumption) isdouble Pareto (Reed, 2001), which has two power law tails around its mode. Toda (2014) argues further that if agents are also heterogeneous when they are born (and the heterogeneity is characterized by a lognormal distribution), then the entire cross-sectional distribution should be
double Pareto-lognormal (dPlN), the product of independent double Pareto and lognormal variables (Reed, 2003).
Why is it of interest to test this particular parametric distribution? There are a few reasons why testing dPlN for consumption is relevant. First, dPlN is as analytically tractable as the lognormal. A property that holds for the log-normal can often be generalized for dPlN. Second, Benhabib et al. (2014) and Toda (2014) have theoretically shown that the size distribution of consump-tion should be approximately lognormal within age cohorts and double Pareto or dPlN in the entire cross-section. Therefore if we find that consumption is dPlN, we have an indirect evidence suggesting that the consumption dynamics obeys the generative mechanism of dPlN. Third, recent techniques for solv-ing heterogeneous-agent models parameterize the cross-sectional distribution to reduce computational complexity (Algan et al., 2008; Reiter, 2009; Winberry, 2015). If dPlN is a good description of reality, it may be useful in computa-tional works because it is flexible, analytically tractable, and has only 3 or 4 parameters.
The main contribution of this note is that I document that dPlN fits re-markably well to the entire cross-sectional distribution of consumption in U.S. dPlN is often not rejected by goodness-of-fit tests, and the lognormal is almost always rejected against dPlN. Therefore consumption is heavy-tailed, and I pro-vide an indirect epro-vidence supporting the recent theoretical models on inequality. Another minor contribution is methodological. Although dPlN has been shown to empirically fit well the size distribution of cities (Reed, 2002; Giesen et al., 2010) and income (Toda, 2012), to the best of my knowledge dPlN has never been formally tested by goodness-of-fit tests in actual data or compared to other parametric distributions except the lognormal. Implementing goodness-of-fit tests requires a lot of coding and computing, and as a consequence, they are rarely applied in practice. Therefore my results may benefit the profession by lowering the hurdle for implementation and disciplining the empirical anal-ysis.3 Given that dPlN is well-grounded by theory and analytically tractable, I
suggest that dPlN should be one of the benchmarks for fitting size distributions.
3See the Online Appendix athttps://sites.google.com/site/aatoda111for details. All
2
A simple generative mechanism of dPlN
In this section, to make the note self-contained, I present a minimal model4that
generates the double Pareto-lognormal distribution and discuss its implications.
2.1
Gibrat’s law and constant birth/death yield dPlN
Consider an economy populated by a continuum of agents indexed byi ∈I = [0,1]. Time is continuous, and letYitbe the income of agentiat timet. Assume
that income obeys Gibrat’s law of proportionate growth. LettingXit = logYit
be the log income, Gibrat’s law (in its simplest form) implies that Xit is a
Brownian motion with drift:
dXit=gdt+vdBit,
wheregis the drift,v >0 is the volatility, andBitis a standard Brownian motion
which is i.i.d. across agents. Assuming that all agents start from Xi0= 0, the
cross-sectional log income distribution at time t is normal: Xit ∼ N(gt, v2t).
Hence the income distribution is lognormal, which is thin-tailed.
To get a fat-tailed distribution, suppose that agents (dynasties) “die” at a constant Poisson rate δ > 0 and are reborn with initial log income x = logy= 0. In steady state, the size of the cohort with ageais e−δa (the density function is exponential, δe−δa), and the log income distribution for this cohort isN(ga, v2a). Therefore the density of the entire cross-sectional distribution is
the normal mixture
Z ∞ 0 1 √ 2πv2ae −(x−ga)2
2v2a δe−δada.
This integral may appear complicated, but actually the integrand has an explicit primitive function (see Toda (2014) for details). The result is
fL(x) = ( αβ α+βe −α|x|, (x≥0) αβ α+βe −β|x|, (x <0) (1)
where−α <0< βare solutions to the quadratic equation v22ζ2−gζ−δ= 0. The
density (1) is known as (asymmetric)Laplace (Kotz et al., 2001). Although the Laplace distribution was derived using a Brownian motion, Section 5 of Toda (2014) shows that it robustly arises as the limit distribution for a large class of stochastic processes. Applying the change of variable x= logy to (1), the density of the income distribution becomes
fdP(y) = ( αβ α+βy −α−1, (y≥1) αβ α+βy β−1, (0≤y <1) (2)
which is known as double Pareto (Reed, 2001).5 As its name suggests, the 4See Reed (2001, 2003), Benhabib et al. (2014), Toda (2014), and Toda and Walsh (2015)
for similar models and Mitzenmacher (2003), Gabaix (2009), and Jones (2015) for reviews.
5Hence the Laplace and the double Pareto distributions have the same relation as the
nor-mal and the lognornor-mal distributions. As an interesting historical remark, Laplace discovered the Laplace distribution in 1774, which predates by a quarter of century the discovery of the normal distribution by himself and Gauss in the early 1800s. For more historical background, see Kotz et al. (2001).
double Pareto distribution (2) has two Pareto (power law) tails, and α, β are called the power law exponents.
So far we assumed that initial log income is zero (agents are ex ante identi-cal), but more generally we may think of situations in which agents are hetero-geneous upon birth. A natural assumption is that initial log income is normally distributed, N(µ, σ2). Then in steady state the log income distribution is the
convolution of the normal and Laplace distributions, which is known as normal-Laplace (Reed and Jorgensen, 2004). The normal-Laplace distribution has four parameters, µ, σ (mean and standard deviation of the lognormal component) andα, β (power law exponents). Through the change of variablex= logy, the income distribution becomes the product of independent double Pareto and log-normal variables, which is calleddouble Pareto-lognormal (dPlN, Reed, 2003).
The double Pareto and lognormal distributions are special cases of dPlN by letting σ→0 andα, β→ ∞, respectively. This is an important point because it means the lognormal distribution, which is nested within dPlN, can be tested against dPlN by the likelihood ratio test. See Reed and Jorgensen (2004) and Hajargasht and Griffiths (2013) for various estimation methods.
2.2
Economic implications of dPlN
The double Pareto-lognormal distribution has a few attractive features. First, while the density of the double Pareto has a cusp at the mode, which does not seem to show up in the cross-sectional data, dPlN is smooth.
Second, since dPlN has a lognormal body and two Pareto tails but contains only 3 (ifα=β) or 4 parameters (ifα6=β), it is flexible yet parsimonious. This point might be convenient for numerically solving heterogeneous-agent models. In these models, the cross-sectional distribution becomes part of an aggregate state, and recent works parameterize this distribution. For example, Algan et al. (2008), Reiter (2009), and Winberry (2015) use exponential families, histograms, and normal mixtures. Since these distributions do not have fat tails but dPlN does, it might be useful in models that match micro data.
Third, because dPlN is the product of independent double Pareto and log-normal variables, which both have closed-form moments, dPlN is analytically tractable. Indeed, if Y is dPlN with parameters (µ, σ, α, β), then Y = Y1Y2
with Y1 double Pareto (with mode 1 and power law exponents α, β) and Y2
lognormal, so itsη-th moment is (see Reed and Jorgensen (2004))
E[Yη] = E[Y1η] E[Y2η] =
( αβ (α−η)(β+η)e µη+1 2σ 2η2 , (−β < η < α) ∞. (otherwise) (3)
Since α, β describe the inequality in the top and bottom of the distribution, we might also use them in applied works on inequality (Kunieda et al., 2014; Garc´ıa-Pe˜nalosa and Turnovsky, 2015) to decompose the inequality in the top and bottom.
This expression has an interesting implication for asset pricing. In the above simple model, since idiosyncratic shocks are multiplicative, the equilibrium is au-tarky (Constantinides and Duffie, 1996) and consumption equals income. From the Euler equation, Balduzzi and Yao (2007) show that the growth rate of the cross-sectional moment of consumption mt+1 = Et+1[ci,t−γ+1]/Et[c−γit ] is a valid
is the relative risk aversion of agents. When consumption is lognormal, so logcit∼N(µt, σ2t), then the SDF simplifies to
mBYt = exp ∆ −γµt+ 1 2γ 2σ2 t , (4)
where ∆Xt = Xt−Xt−1. When consumption is dPlN, using (3) for Y =
cit, ci,t−1 andη=−γ, the SDF becomes
mdPlNt = exp ∆ −log(1 +γ/αt)−log(1−γ/βt)−γµt+ 1 2γ 2σ2 t . (5) Clearly (4) is a special case of (5) by lettingαt, βt→ ∞. In each case the SDF
is a simple expression of the parameters. One caveat is that we needγ < βtfor
moments of dPlN to exist.
3
Testing dPlN for consumption
In this section I estimate and test dPlN using U.S. consumption data. I use the same data as the real, seasonally adjusted, quarterly household consump-tion data from the Consumpconsump-tion Expenditure Survey (CEX) used in Toda and Walsh (2015). The data consists of 410,788 observations from December 1979 to February 2004. Since households are surveyed every three months but in different months, on average we have about 1,400 households in each month (or 4,200 in each quarter).
3.1
Statistical model and parameter estimation
In the simple model of Section 2, we found that the cross-sectional distribution of consumption is dPlN. If log consumption has a time trend but the shocks have constant volatility, then the location parameterµwill be time-dependent but σ, α, β will be constant. It is well known that the consumption data in the CEX is subject to substantial measurement error, but dPlN survives if (the multiplicative) measurement error is lognormal because dPlN already has a lognormal component.
Now suppose that observed consumption is dPlN with parameters (µt, σ, α, β)
(or observed log consumption is normal-Laplace with the same parameters). Normalizing log consumption by subtracting the population mean, normalized log consumption is normal-Laplace with parameters (µ, σ, α, β) (whereµis such that the mean is zero, so µ = 1
β −
1
α), which do not depend on the sample.
Since the CEX samples the same households once in a quarter, by the above reasoning monthly data of normalized consumption has the same distribution as thequarterlydata of normalized consumption obtained by pooling three con-secutive monthly data, which contains no overlapping households. Making the sample size approximately three times larger in this way, I am taking a con-servative position since it is easier to reject a particular parametric model with more data.6
6Of course, one may object pooling different data sets (although here it is theoretically
justified). As a robustness check, I also perform all subsequent analysis with the original monthly data in the Online Appendix but the results are similar.
Having constructed quarterly normalized log consumption data, I estimate the normal-Laplace parameters (µ, σ, α, β) for each quarter by maximum likeli-hood. Since the two power law exponents α, βare almost the same, I estimate the parameters of the symmetric normal-Laplace distribution by maximum like-lihood. The likelihood ratio test failed to reject symmetry (α =β) in 77 out of 98 quarters at significance level 0.05. (The Online Appendix verifies through simulations that the likelihood ratio test has high power and hence the non-rejection of symmetry is not due to low power.) Therefore I choose the sym-metric normal-Laplace distribution as the benchmark model for normalized log consumption.
Figure 1(a) shows the histogram of log consumption for 1985:Q1, together with the fitted symmetric normal-Laplace density plotted in the range between the minimum and the maximum log consumption (other quarters look similar). We can see that the fit is quite good. Figure 1(b) shows the maximum likelihood estimate of the power law exponentαfor each quarter. The power law exponent is around 4 (the average across all quarters was 4.06) and in the range [3,5] except 1993:Q4, whereαb= 5.5. The sample average of the standard error across all quarters was 0.31. These results are similar to Toda and Walsh (2015).
-2 -1 0 1 2 frequency 0 100 200 300 400
(a) Histogram and fitted density for 1985:Q1
year
1980 1985 1990 1995 2000 2005
power law exponent
2.5 3 3.5 4 4.5 5 5.5 6
(b) Power law exponentα
year
1980 1985 1990 1995 2000 2005
log variance parameter
0.3 0.35 0.4 0.45 0.5
(c) Log variance parameterσ
log variance parameter
0.3 0.35 0.4 0.45 0.5
power law exponent
2.5 3 3.5 4 4.5 5 5.5 6 (d) Scatter plot ofσ, α
Figure 1: Maximum likelihood estimation of symmetric normal-Laplace distri-bution to quarterly U.S. normalized log consumption data.
What is the meaning of a power law exponent of 4? For income, it is well-known that the exponent is about 2 or 3 (Toda, 2012). Therefore the
simple model in Section 2—which predicts the same exponent for income and consumption—is clearly false. Thus an interesting question might be to look at the dynamics of household income and consumption and investigate how we should modify the model in order to explain the tail heaviness of income and consumption observed in the data, but is left for future research.
Figure 1(c) shows the estimated log variance parameterσfor each quarter, which shows a similar pattern to Figure 1(b). Indeed, the scatter plot ofσ,b αb in Figure 1(d) shows a positive relation (correlation 0.81). However, this is probably an artifact of sampling error. For dPlN, tails are fatter when σ is large or α is small. Hence for a fixed sample, there is a trade-off in the fit between increasingσand decreasingα. At the best fit parameter values, when one is large, the other also tends to be large. Indeed, I estimated the parameters from 100 simulated datasets with sample sizeN= 4,000 and dPlN parameters (µ, σ, α, β) = (0,0.4,4,4), and obtained similar graphs and correlation 0.89.
How economically important is it to model consumption as dPlN instead of lognormal? Figure 2 shows the Balduzzi-Yao (BY) stochastic discount factor in (4) and the dPlN SDF in (5) for relative risk aversion γ= 1 and 2.5. With low risk aversion (γ = 1, Figure 2(a)), the two SDFs are virtually identical (correlation 0.998). However, even slightly increasing the risk aversion (γ= 2.5, Figure 2(b)) makes the two SDFs quite different (correlation 0.85): the dPlN SDF is more volatile. year 1980 1985 1990 1995 2000 2005 SDF 0.8 0.9 1 1.1 1.2 1.3 dPlN BY (a)γ= 1 year 1980 1985 1990 1995 2000 2005 SDF 0.6 0.8 1 1.2 1.4 1.6 1.8 dPlN BY (b)γ= 2.5
Figure 2: dPlN and BY stochastic discount factor.
3.2
Goodness-of-fit
According to Figure 1(a), the symmetric normal-Laplace distribution seems to fit the log consumption data well. We can also see that there are large positive and negative values that would be unlikely if the distribution were normal. To see this visually, Figure 3 shows the quantile-quantile plot (QQ plot) of log consumption against the fitted normal (Figure 3(a)) and normal-Laplace (Figure 3(b)). In Figure 3(a), the actual quantiles (vertical axis) are more extreme than those of the normal distribution (horizontal axis), suggesting that the log consumption distribution has heavier tails than the normal distribution. On the other hand, the QQ plot against the normal-Laplace distribution shows a virtually straight line, suggesting the good fit.
Quantiles of fitted normal -2 -1 0 1 2 Quantiles of data -2 -1 0 1 2 (a) Normal
Quantiles of fitted normal-Laplace
-2 -1 0 1 2 Quantiles of data -2 -1 0 1 2 (b) Normal-Laplace
Figure 3: Quantile-quantile plot of normalized log consumption. 1985:Q1.
Although a picture is worth a thousand words, to evaluate the goodness-of-fit of the double Pareto-lognormal distribution (withα=β) and the lognormal distribution more formally, I perform both the Kolmogorov-Smirnov (KS) test (Massey, 1951) and the Anderson-Darling (AD) test (Anderson and Darling, 1952) and compute the P value by bootstrapping 500 times for each quarter as explained in the Online Appendix. LettingF(x) be the theoretical distribution andFN(x) the empirical cumulative distribution function, the KS and AD test
statistics are based on the sup andL2 norms,
sup x |FN(x)−F(x)| and Z ∞ −∞ (FN(x)−F(x))2 F(x)(1−F(x)) dF(x).
Note that the KS test has a low power for detecting deviations from the theo-retical distribution in the tails becauseFN(x)−F(x) tends to zero asx→ ±∞.
On the other hand, the AD test can detect deviations in the tails because the weighting function [F(x)(1−F(x))]−1 tends to infinity as x → ±∞. Hence,
with the Anderson-Darling test the deviations in the tails are more penalized. (The Online Appendix verifies through simulations that the KS test has indeed a low power.)
Since I am interested in the tail heaviness, for my purpose clearly the AD test is more appropriate. However, I also perform the KS test because it is widely used.7 The double Pareto-lognormal distribution (with α = β) is not rejected by the KS test at significance level 0.05 in 79 quarters out of 98. On the other hand, the lognormal distribution is rejected in 73 quarters. When using the AD test, which is more relevant because I am interested in the tail behavior, dPlN is not rejected in 64 quarters out of 98, whereas lognormal is rejected in 92 quarters. The fact that the lognormal distribution is rejected more often by the AD test suggests that the lognormal distribution fails to fit the tails of the
7In testing the lognormal distribution within age cohorts, Battistin et al. (2009) perform
the KS test as well as tests based on skewness measures that use quantiles (but not the AD test). Since their skewness measures use only octiles (12.5 percentile, 25 percentile, etc.), their test cannot address the fit in the tails (1 percentile or 99 percentile, say). In fact, as I show below in testing lognormality against dPlN, within age cohorts the KS and AD tests reject the lognormal distribution about 30% and 50% of the time, which suggests that the tails indeed matter for evaluating the model fit.
data, which is consistent with the QQ plot in Figure 3(a). The Online Appendix shows the actual P value of these tests.
3.3
Comparison to other parametric distributions
I also compare the performance of dPlN to other parametric distributions with different tail heaviness using the Bayesian Information Criterion (BIC). (Using AIC gives similar results.) The parametric distributions that I consider are the lognormal, gamma, and generalized beta II (GB2) of McDonald (1984).8 GB2
has four parametersa, b, p, q with density fGB2(x) =
axap−1
bapB(p, q)(1 + (x/b)a)p+q,
where b >0 is a scale parameter,a, p, q >0 are shape parameters, andB(p, q) denotes the beta function. The attractive feature of the generalized beta II distribution is that it nests a wide range of parametric distributions such as the exponential, gamma, lognormal, Weibull, and other distributions as special or limiting cases.9 Out of 98 quarters, dPlN performed best in 78 quarters,
GB2 in 15 quarters, lognormal in 5 quarters, and gamma in none. Therefore, among a large class of parametric distributions, dPlN provides the best fit to the consumption distribution.
3.4
Testing dPlN against lognormal distribution
Since the lognormal distribution is nested within the double Pareto-lognormal distribution (by letting the power law exponents α, β to infinity), we can test the lognormal distribution against dPlN by the likelihood ratio test. The test rejects the lognormal distribution at significance level 0.05 in every quarter except 1993:Q4, with P value 0.08 (which gives the largest power law exponent 5.5 in Figure 1(b)). Therefore the consumption distribution is better described by dPlN than lognormal when we look at the entire sample.
This finding does not necessarily contradict to those of Battistin et al. (2009) because they look at the consumption distribution within age cohorts, not the entire cross-section. Since the double power law emerges from the constant prob-ability of the “birth/death” process as in the model in Section 2, we would expect that the cross-sectional consumption distribution is more lognormal within age cohorts than in the entire cross-section. To evaluate this conjecture, I perform the likelihood ratio test and goodness-of-fit tests for the lognormal distribution for each age cohort. The groups are household head age 30 or less, 31 to 40, 41 to 50, 51 to 60, and 60 or more. The likelihood ratio test fails to reject lognormality in 46, 38, 37, 56, and 32 quarters out of 98 for each age group, respectively. The Kolmogorov-Smirnov test fails to reject lognormality in 67, 69, 63, 72, and 43 quarters for each age group, and the Anderson-Darling test fails to reject in 51, 51, 50, 64, and 23 quarters, respectively. Therefore the lognormal distribution fits reasonably well to the cross-sectional distribution of
8I do not compare to the classic L´evy alpha-stable distribution (Mandelbrot, 1961) because
(i) it does not have a closed-form density, and (ii) its power law exponent is necessarily below 2, whereas for consumption the exponent is well above 2.
9See McDonald (1984) and Toda (2012) for more discussion. GB2 does not nest the double
consumption for each age group, in agreement with Battistin et al. (2009). My finding that the double power law emerges only in the entire cross-section and not within age cohorts is consistent with the theoretical model of Benhabib et al. (2014) and Toda (2014), in which the age distribution is geometric due to the constant probability of birth/death.
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